Monotonic penalized-likelihood image for X-ray fluorescence computed tomography

In this work, we derive a monotonic penalized-likelihood algorithm for image reconstruction in X-ray fluorescence CT (XFCT) when the attenuation maps at the energies of the fluorescence X-rays are unknown. XFCT is not a transmission tomography modality but rather a stimulated emission tomography modality, and it is thus necessary to correct for attenuation of the incident and fluorescence photons if accurate images are to be obtained. This is challenging because the attenuation map is, in general, known only at the stimulating beam energy and not at the various fluorescence energies. We have developed a penalized-likelihood image reconstruction strategy for this problem. The approach alternates between updating the distribution of a given element and updating the attenuation map for that element´s fluorescence X-rays. As derived, the approach is guaranteed to increase the penalized likelihood at each iteration.